Uses of Zeno’s “Achilles Paradox” in Historical Geology
Rossbach, Thomas J
The Greek philosopher Zeno put forward a paradox that Achilles would never be able to overtake a tortoise in a foot race if the tortoise was given a head start. The Achilles paradox is an example of the summation of an infinite series that always approaches, but never reaches, unity. A similar paradox can occur when teaching students about intermediate forms between any two taxa in an evolutionary lineage. While the mathematical solution to the paradox is simple enough, the concept of the paradox plays a role in comparing gradual and punctuated modes of evolution. The discovery of a single transitional form creates the need for two more transitional forms (for a total of three intermediaries), which then requires the discovery of four more intermediaries (for a total of seven) and so on. Similar to the paradox, it appears that an infinite number of intermediaries must be found to connect any two taxa. The Achilles paradox has other applications in historical geology, such as radiometric decay. Showing how the paradox relates to several aspects of science may encourage better critical thinking on the part of students.
Zeno of Elea (c.490-c.420 BCE) was a preeminent member of the Eleatic school of philosophy, which denied the usefulness of the senses as a means of attaining truth. The Eleatics attempted to show that by using reasoned arguments they could ignore what their senses might tell them. Zeno presented Greek philosophers with four famous paradoxes, all of which seemed to disprove the possibility of motion as it was perceived. All of these paradoxes were based on the assumption that space and time are infinitely divisible. The most well known of these is the Achilles paradox, which will be discussed in detail below. Like many philosophers, Zeno set forth these paradoxes to stimulate debate and, according to other interpretations, to show the incompleteness of mathematical understanding as it stood at that time (Asimov, 1976).
Today we recognize the Achilles paradox as dealing with a convergent series of infinite terms that never seems to arrive at a finite answer. However, nearly 2,100 years after Zeno’s death, mathematician James Gregory (1638-1675) demonstrated that converging series exist in which an infinite number of terms do add up to a finite sum (Asimov, 1976). Applications and variations on the Achilles paradox can be used in certain aspects of historical geology, especially with evolutionary lineages and radiometric decay.
THE ACHILLES PARADOX
Achilles: “A footrace? How outrageous! Me, the fleetest of all mortals, versus you, the ploddingest of all plodders! There can be no point to such a race.”
Tortoise: “You might give me a head start.”
Achilles: “But I will catch you sooner or later-most likely sooner.”
Tortoise: “Not if things go according to Zeno’s paradox, you won’t.”
The exchange above from Hofstadter (1979) succinctly sets up the Achilles paradox. Achilles and a tortoise are scheduled to compete in a footrace. The tortoise, being the slower of the two, asks for a head start in order to make the race more sporting. Achilles agrees, feeling confident that he can beat his slow-footed opponent no matter how much of a head start the tortoise is given. As the race begins (Figure 1), Achilles is at the starting line (A1) and the tortoise is at his head start position (T1). In a very short time, Achilles makes up this distance arriving at the tortoise’s original position (A2). But during this time, the tortoise has moved ahead (T2). Achilles makes up this distance (A3) but, again, the tortoise moves ahead (T3) during the time it took for Achilles to cover the distance. According to Zeno, Achilles will never be able to catch up to and overtake the tortoise because the tortoise always moves ahead during the time it takes Achilles to reach the tortoise’s previous position. The outcome is that the tortoise will reach the finish line (T4) just ahead of Achilles (A4).
Since most classrooms are lacking ancient Greek athletes, if not tortoises, a more practical demonstration of the Achilles paradox is needed. The simplest involves an ordinary pencil, a tabletop, and the force of gravity. Hold a pencil a few feet above a convenient tabletop and ask “What will happen if I release the pencil?” Students will likely answer “It will fall and hit the table” (if they only respond with “It will fall,” prompt them by asking “How far?”). Without letting go of the pencil, announce that you can prove the released pencil will never hit the table. Explain that before the pencil can travel the entire distance from your hand to the table (D) it will have to fall half that distance (0.5 D) leaving 0.5 D remaining to fall. Then, before it can travel the remaining distance, it will have to cover half that remaining distance (0.25 D), and then half the subsequent remaining distance (0.125 D), and so on. You can show that the pencil will have to cover an infinite number of half-distances on its way down to the table. At an extreme, the pencil is at a distance of one hydrogen atom above the table, but the pencil will still have to cover half the atom’s diameter, then half of that distance, then half of that distance, and so on. Like Achilles never quite catching up to the tortoise, it seems that the pencil will never quite reach the table. At this point, release the pencil, watch it plummet down and resoundingly hit the table. Ask the students why the pencil actually made it to the table, even though it had to traverse an infinite number of half-distances.
THE PARADOX, THE FALLACY, AND ITS SOLUTION
At first, the pencil problem appears to be one of a summing of values, each of wruch is one-half of the previous value; i.e., 1/2 + 1/4 + 1/8 + 1/16 etc. This is represented by the summation equation shown in Figure 2. At n=10, the value reaches 0.999. All higher values of n produce a result above 0.999, but still never reaching unity. Therefore, it seems that the pencil will never reach the table. We know this is false because the pencil will hit the table when released.
The fallacy is that although the number of half-distances is infinite, the total amount of time needed to traverse those distances is not necessarily infinite. For this example we can use a simple acceleration-time-distance formula, D= (½)(A)(T^sup 2^). Since the total distance is known (height the pencil is held above the table), the acceleration is known (acceleration due to gravity), the amount of time during which the pencil will fall is also known [T = (2D/A)^sup 0.5^] and is therefore finite.
The solution is found in realizing that what is being dealt with in the paradox is the summation of an infinite feometric series (Sn) with a common ratio (Gullberg, 997), which is usually encountered in the section of calculus dealing with infinite series. An infinite geometric series is convergent if the common ratio lies strictly between -1 and +1. In the pencil example, the common ratio is ½ As seen in Figure 3, the sum of an infinite, convergent geometric series where the first term in ½ and the common ratio is ½ is 1, and the pencil hits the table.
“Infinity torments me.” – Alfred de Musset (1810-1857)
“Neither is there a smallest part of what is small, but there is always a smaller, for it is impossible that what exists should cease to be.” – Anaxagoras (500? – 428? BCE)
My interest in applications of the Achilles paradox began in 1984 when I was a first-year graduate student at UNC-Chapel Hill taking a course in micropaleontology from Dr. Joseph St. Jean. I was asked on an exam to compare gradualism with punctuated equilibrium regarding me evolution of Foraminifera. While writing out my answer I was struck by an odd thought: “Could there be gradualism in the evolutionary lineage of a single-celled organism?” From my understanding at the time, gradualism called for intermediary forms between any two given taxa, so A led to B which led to C; but then intermediaries would be needed between A and B and between B and C. Extrapolation of this led to the problem of an infinite number of intermediates. The punctuated equilibrium model of Eldredge and Gould (1972) attempted to resolve this conundrum by having a given morphology remain stable for a given amount of geologic time, then evolve into the new form in a geological instant. In that sense, one could go from A to C without having to develop B in between. This “quantum” view of evolution was very appealing for those stymied by a lack of intermediaries, or for those tired of looking for them. See Gould (2002) for a more detailed explanation of the model.
For the Foraminifera, gradualism did not seem to make sense. If you were dealing with a dinosaur, one could argue that a slight change in tooth shape, skull length or claw size represents intermediate forms as they represent changes in only certain aspects of the organism (or course, you are actually changing the entire organism with each new development). But with Foraminifera, consisting of only one cell, any change by definition changed the entire organism regardless of semantics. From this point of view, all morphologic change within a foram lineage has to be punctuated rather than gradualistic. Hence, my exam answer stated Gradualism does not exist for Foraminifera.” (I have forgotten what Dr. St. Jean said about my answer, but I passed the course.) From that time forward, the contrast between gradualism and punctuated equilibrium has always fascinated me.
This variation of the Achilles paradox deals with the concept of “missing links” within an evolutionary lineage. In Figure 4A, taxon X changes morphology over time to become recognized as taxon Y. The time scale is not relevant; only the concept that time is required for the change is.
In a pure gradualistic approach, an intermediate form is required to bridge the morphologic gap between X and Y. This first intermediary is represented by the black dot in Figure 4B. However, the recognition of this intermediary then produces a need for two more intermediaries, one between it and X and one between it and Y. These additional intermediaries are shown by the striped dots in Figure 4C. There are now three intermediaries between X and Y. But now there is a need for additional intermediaries to fill the gaps made by the addition of the previous dots. These intermediaries are shown by the stippled dots in Figure 4D, now making a total of seven intermediaries between X and Y. Each step requires a total of 2^sup n^ -1 intermediaries. It does not take long for the number of “missing links” to become unwieldy. At the tenth iteration the number of intermediaries would total 2^sup 10^ -1, which equals 1,023. Similar to the pencil example, it seems that an infinite number of intermediate steps (or “missing links”) are required to bridge the gap between taxa X and Y. Again, we know this cannot exist because we cannot have an infinite number of individuals or populations acting as intermediaries. The time span between an individual’s birth and its ability to reproduce cannot be infinitesimally small. Since the time span between X and Y is finite, the number of intermediaries must also be finite.
The realization that there cannot be an infinite number of intermediaries helps to answer one common argument against evolution, that of so-called missing links. If the argument is made that evolution cannot provide all possible intermediaries, one can show that an infinite number of intermediaries cannot exist. If there cannot be an infinite number of intermediaries, then there must be a finite number of intermediaries (whether or not they will all be found due to the vagaries of fossil preservation or accessibility to the rocks is another matter), so simply negating evolutionary lineages due to a perceived lack of every possible intermediate form is an untenable position. The argument is similar to saying that since I am 5’10” tall and my father is 6’0”, I cannot be descended directly from my father because there is no intermediary who is 5’11”.
The point being made is that a change in morphology from one taxon to another will not allow for all possible intermediate forms, and that even in a gradualistic approach the finite number of intermediaries will by necessity eventually call for a “quantum” change without any additional intermediaries. In other words, a short, punctuated jump would fit in between any two dots in Figure 4, although looking at it over a long time frame would make this jump too small to discern and you would be left with the appearance of a straight-line gradualistic model. Even within relatively well-documented lineages like the horse, the route from Hyracotherium to Equus, while seemingly dotted with intermediary forms (Miohippus, Merychippus, etc.), is only gradualistic when seen from a distance. There are not an infinite number of genera/species between the two end-members. The evolutionary space between any two taxa, no matter how apparently small, must seem to be punctuated at some level. If this notion is dismissed, it seems that the idea of an infinite number of intermediaries would creep back in. Gould (1994), in discussing the evolution of complex organisms between 600 and 530 million years ago, makes the point that the development of complexity is discontinuous and episodic, not gradually accumulative.
The Achilles paradox also has applications in teaching radiometric decay. Similar to the constantly shrinking distances Achilles covers without reaching the tortoise, the constant halving of parent atoms in the decay process could be seen as never allowing the final parent atom to completely disappear.
Figure 5 shows a typical graphic representation of the decay of parent isotope atoms (black dots) into the daughter product (white dots). During each successive half-life (T½), half of the previously existing parent atoms decay into the daughter product. In Figure 5, we start with 32 parent atoms at T½ = 0, with 16 parent atoms remaining when the first half-life is reached, 8 by the second half-life, 4 by the third, 2 by the fourth, and 1 remaining parent at the end of the fifth half-life.
It is at this point that the paradox appears. If we say that during each half-life one-half of the parent atoms will decay, what happens when only one parent atom is left? By the strict definition, there will be one-half of a parent atom left after the next half-life, one-quarter of a parent atom after the next, one-eighth after the next, and so on. Similar to the pencil never reaching the table, that final parent atom never seems to completely vanish. At the risk of sounding facetious, I have indeed had students ask me if this would happen when only one parent atom is left.
There are several ways of dealing with the paradox. The simplest is to explain that you cannot have half an atom of a given element. For example, if you had a single unstable ^sup 14^C atom left and by some process you could get it to split perfectly in half, the resulting pieces could not even be carbon (atomic number 6), but instead would be two atoms of lithium (atomic number 3); so ^sup 14^C[arrow right] 2 ^sup 7^Li. The fact that this is not how ^sup 14^C decays also helps. Similarly, a single remaining ^sup 238^U atom will not decay into two atoms of ^sup 119^Pd. Potassium-Argon decay would show another reason why this cannot happen. Since potassium is atomic number 19, if it were to split perfectly in half the resulting two atoms would both be of atomic number 9.5. Not even quantum physics will allow for half a proton.
The general solution to the paradox tells us that the final parent atom will decay, but the statistics do not tell us exactly when the atom will decay. From a statistical point of view, any parent atom has a one in two (1:2) chance of decaying during any given half-life interval. In the example above, we Know that of the 32 original parent atoms it is most likely that one-half (or 16) will decay during the interval between 0 and the first half-life, but we cannot predict which 16 will decay. Each of the original 32 atoms has a 1:2 chance of decaying during the first interval. Similarly, each of the remaining 16 parent atoms has a 1:2 chance of decaying during the second half-life interval. By the fifth half-life, only one parent atom remains. While we predict that only one parent atom will remain by this time, the probability of that particular atom surviving until this time is quite small [(½)^sup 5^ = 0.03125, or 1:32). All that can be said is that the probability of that single atom remaining until the sixth half-life interval is 1:2, until the seventh is 1:4, until the eighth is 1:8, etc. We cannot actually predict when that last atom will decay, only that as time goes on the chances of its continued existence diminishes.
If during each half-life exactly one-half of the parent atoms decay, the number of half-lives required for a single parent atom to remain is (log n ÷ log 2); where n = the total number of parent atoms. In the example above, we started out with 32 parent atoms, so (log 32) ÷ (log 2) = 5, meaning that it required five half-lives to get down to one remaining atom. Variations on this mathematical relationship may help students in introductory geology classes become more familiar with the half-life concept (and with the powers of 2).
Asimov (1957) puts forth some interesting problems that deal with the naif-life concept. For example, start with 1,048,576 atoms of isotope X that decays with a half-life of one day. How long will it take to get down to one atom of X? First have the students divide 1,048,576 by 2 until they reach a value of 1. This will take twenty steps; hence it will take 20 days to reach a single atom of X. Then show them the quick way: (log 1,048,576) ÷ (log 2) = 20.
Now expand on this premise. Assume that the entire Earth is composed of isotope X. Ask students how long it would take until there was only one atom of X left. After they have made their guesses, give them the number of atoms that make up the Earth (approximately 1.33×10^sup 50^) (Weisenberger, 2004). The calculation gives an answer of: (log 1.33×10^sup 50^) ÷ (log 2) [approximate] 167 days, or approximately 5.6 months. With this information, ask them how long it would take to get down to one atom if the entire galaxy were composed of X. After their best guesses, give them the number of atoms in our galaxy (approximately 5×10^sup 68^) (Champion, 1998). The calculation gives an answer of (log 5×10^sup 68^) ÷ (log 2) [approximate] 228 days, or approximately 7.6 montns. Now assume that the entire universe was composed of X. Again, have them make a best guess. Using a value of 1×10^sup 79^ for the number of atoms in the universe (Radick, 1998), the calculation results in an answer of (log 1×10^sup 79^) ÷ (log 2) [approximate] 262 days, or approximately 8.7 months. Variations on this theme are numerous, such as using the number of atoms in a human body or the number of atoms in their textbook, or calculating how long it should take a single gram of pure ^sup 238^U (2.53×10^sup 21^ atoms) to decay down to a single remaining ^sup 238^U atom (approximately 71.1 half-lives at 4.5 billion years per half-life or 320 billion years).
The Achilles paradox appears to be a simple puzzle, but freat scientific insights are based on its solution. It has een suggested that the atomic theory of Democritus (C.470-C.380 BCE), which put forward that matter consisted of exceedingly small, but finite, indivisible particles (rather than the notion that matter could be ground down into infinitely finer particles) was inspired y Zeno’s paradoxes. From Democritus grew modern atomic theory, first set forth by chemist John Dalton (1766-1884).
The idea that energy, like matter, is not infinitely divisible but exists in discrete packets, the cornerstone of the quantum theory of physicist Max Planck (1858-1947), conceivably could also be traced back to Zeno. Even the calculus of Isaac Newton (1642-1727) and Gottfried Leibnitz (1646-1716) is partly based on solving for an infinite number of subdivisions under a curve. In biology, Gregor Mendel (1822-1884) demonstrated that inherited characteristics remained distinct, rather than being infinitely blended down the generations. Again, shades of Zeno.
It is possible that life (as we define it) may also be affected by the paradox. Gould (1994) states: “For reasons related to the chemistry of life’s origin and the physics of self-organization, the first living things arose at the lower limit of life’s conceivable, preservable complexity.” This could be restated as: “There is a definite smallest size at which life can exist, below which metabolic functions cannot proceed.” Part of the recent debate over whether meteorite ALH 84001 held Martian fossils deals with this point. The alleged fossils are extraordinarily small, each measuring 20 to100 nanometers across and have only one-thousandth the volume of a small terrestrial bacterium. Many biologists believed that nothing this small could hold the biochemical machinery needed to sustain a replicating organism (see Darling, 2000), but living terrestrial nanobacteria within this size range are being reported (though questions about whether they contain DNA or replicate through inorganic microcrystallization are still being investigated). So while the lower limit of life is still being derated and sought after, it seems logical that there must be a lower size limit below which true metabolic functions cease to operate.
While Zeno’s Achilles paradox has had a long history in chemistry, physics and biology, its applications are also relevant in teaching geology, especially in explaining evolutionary lineages and radiometric decay. The concept that infinite divisibility may be self-defeating might encourage better critical thinking on the part of the student when faced with similar problems dealing with infinite series.
I wish to thank my brother, Andrew A. Rossbach, for his assistance in checking the mathematics and for other useful suggestions in the preparation of this paper.
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Gould, S. J., 1994, The evolution of life on Earth, Scientific American, v. 271, p. 85-91. Reprinted 2004, Dinosaurs and Other Monsters. Scientific American Special Edition, v. 14, p. 92-100.
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Radick, R., 1998, The universe, http://www.sunspot .noao.edu/ sunspot/ pr/ answerbook.html.
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Thomas J. Rossbach Department of Geological, Environmental and Marine Sciences, Elizabeth City State University, Elizabeth City, NC 27909 email@example.com
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